CAS ANNUAL MEETING, NOVEMBER 2009 Dan Murphy, FCAS, MAAA Trinostics - - PowerPoint PPT Presentation

INTRODUCTION TO RESERVE RANGE THEORY AND PRACTICAL MODEL APPLICATION CAS ANNUAL MEETING, NOVEMBER 2009 Dan Murphy, FCAS, MAAA Trinostics LLC

Agenda

• Motivation
• Terminology
• Popular stochastic techniques

– Mack – Monte carlo simulation – Bootstrapping

• Aggregation of liabilities

CAS Annual Meeting 2009

2

Why measure ranges?

• NAIC

– “We know carried reserves can’t be perfectly omniscient. We’ll settle for reasonable, with justification.”

• Rating Agencies

– Looking for ways to objectify rating – Moving from reserve adequacy to economic capital

• Fair Value Accounting

– Value of an asset recognizes uncertainty of future cash flows – Concept being applied to liabilities

• Economic Capital

– Sufficient capital to be 99.5% sure that balance sheet entries will not change over the next year by amounts large enough to ruin the firm

• Solvency II Solvency Capital Requirement (SCR)
• Transparency

– If Wall Street understood our company better maybe we’d get a better rating

CAS Annual Meeting 2009

3

Practical reasons to provide ranges

• People think we already do. We’re the math

geeks after all!

• If actuaries don’t, somebody else will
• Knowing uncertainty of an estimate can

improve decisions based on that estimate

CAS Annual Meeting 2009

4

A practical situation where knowing a range can help

Home to airport via back roads Average = 43 minutes

CAS Annual Meeting 2009

5

Hmm, if I took the freeway I could get in a power nap

Home to airport via freeway Average = 32 minutes (per google maps)

CAS Annual Meeting 2009

6

Home to airport via freeway “With traffic add 20-30 minutes”

Do I risk being late for you-know-who or take sure bet?

CAS Annual Meeting 2009

7

“Risk comes from not knowing what you’re doing.”

• Warren Buffet

CAS Annual Meeting 2009

8

Top 5 List for Not Giving a Range

Attendance at session required to see list! 

CAS Annual Meeting 2009

9

SSAP 55 vs. GAAP: Who gave accountants all that say anyway?

• SSAP 55 Effective 2001

– Management’s estimate

• Management shall record its “best estimate”

– Ranges of estimates

• Management may consider a range of reserve estimates
• The range shall not include the set of all possible outcomes but only

those outcomes that are considered reasonable

• When no estimate within the range is better than any other, the

midpoint of the range is to be accrued

• When the high end of the range cannot be quantified, management’s

best estimate shall be recorded

• GAAP

– When a range of estimates exists and no estimate is better than any other, the company shall accrue the lowest estimate in the range CAS Annual Meeting 2009

10

Actuarial standards regarding “ranges”

• riginally couched in terms of actuarial methods

ASOP 36 (2000): Statements of Actuarial Opinion Regarding Property/Casualty Loss and Loss Adjustment Expense Reserves

• Company’s stated reserve amount should be within the

actuary’s range of reasonable reserve estimates

• A range of reasonable estimates is a range of estimates

that could be produced by appropriate actuarial methods or alternative sets of assumptions that the actuary judges to be reasonable

• The reasonable range need not be disclosed

CAS Annual Meeting 2009

11

ASOP “range” wording is evolving: becoming broader, more mathematical

ASOP 43 (2007): Property/Casualty Unpaid Claim Estimates

• One should consider uncertainty associated with one’s

estimate

• Sources of uncertainty may include model risk,

parameter risk, and process risk

• If a range is specified, its basis should be disclosed, e.g.,

– Based on individual estimates, each of which is a reasonable estimate on a stand-alone basis – A confidence interval produced by a model or models – A confidence interval reflecting certain risks, such as process risk and parameter risk

CAS Annual Meeting 2009

12

• What motivates the model behind the Mack methodology?
• How can the calculations be done in a spreadsheet?
• References

– Mack, “Distribution Free …,” Astin 1993, http://www.casact.org/library/astin/vol23no2/213.pdf – Murphy, “Unbiased LDFs,” PCAS 1994, http://www.casact.org/pubs/proceed/proceed94/94154.pdf – Bardis, Majidi, Murphy, “Flexible Factor Chain Ladder Model,” summer eForum 2009, http://www.casact.org/pubs/forum/09sumforum/01_Murphy.pdf – Barnett, Zehnwirth, “Best Estimates for Reserves,” PCAS 2000, http://www.casact.org/pubs/proceed/proceed00/00245.pdf

Excel-erate Your Mack Method

CAS Annual Meeting 2009

13

Does historical variability have anything to say about future variability in a chain ladder application?

• Chain ladder estimate of ultimate loss calculated by squaring the

triangle rather than by vector multiplication of diagonal and LDFs

• Variance of chain ladder estimate will also be calculated by squaring
• Start by looking at first future diagonal

ABC Insurance Company Chain Ladder Loss Projection

AY \ Age 1 2 3 4 5 6 7 8 9 = Ult 2000 10,238 24,654 38,025 46,550 52,842 58,722 65,227 67,604 69,559 2001 5,508 16,235 25,586 32,863 38,111 42,315 45,171 47,666 49,045 2002 7,374 20,620 34,220 43,438 50,898 55,475 58,367 60,943 62,706 2003 6,153 19,182 31,005 40,424 46,949 50,942 54,931 57,354 59,014 2004 7,253 25,066 40,134 51,063 58,376 64,144 69,166 72,218 74,307 2005 10,855 38,520 62,348 82,710 95,382 104,806 113,011 117,998 121,411 2006 10,313 34,341 51,110 65,632 75,688 83,166 89,677 93,634 96,343 2007 16,411 42,228 66,770 85,743 98,879 108,649 117,155 122,324 125,863 2008 21,234 63,281 100,059 128,491 148,177 162,818 175,564 183,311 188,614 All Yr Wtd 2.980 1.581 1.284 1.153 1.099 1.078 1.044 1.029 Simple Avg 3.022 1.586 1.280 1.154 1.099 1.077 1.046 1.029

CAS Annual Meeting 2009

14

Visualization of age 1-2 development suggests the model

• First term bX expresses expected value of linear relationship

– Intercept in more general Y=a+bX does not appear necessary

• Second term expresses random deviations from expected

– Form of z unspecified (“Distribution Free”) but should be symmetric – Heteroscedasticity: higher value of X → higher variability of Y

• Because of square root, optimal value of b that minimizes the sum of

squared residuals (“least squares”) is 2.980

• Estimates of b and σ can be calculated by Excel’s LINEST function

z X bX Y  + = z X 

2000 2001 2005 2007

2008

75000 25000

Y = Age 2 Loss

X = Age 1 Loss

AY \ Age 1 2 2000 10,238 24,654 2001 5,508 16,235 2002 7,374 20,620 2003 6,153 19,182 2004 7,253 25,066 2005 10,855 38,520 2006 10,313 34,341 2007 16,411 42,228 2008 21,234 63,281 All Yr Wtd 2.980

CAS Annual Meeting 2009

15

A B C D E F G 1 AY \ Age 1 2 ata 2 2000 10,238 24,654 2.408 3 2001 5,508 16,235 2.948 4 2002 7,374 20,620 2.796 5 2003 6,153 19,182 3.118 6 2004 7,253 25,066 3.456 7 2005 10,855 38,520 3.549 8 2006 10,313 34,341 3.330 9 2007 16,411 42,228 2.573 10

sum / wtd avg 74,105 220,845

2.980 11 12 b 2.980 a 13 se(b ) 0.157 #N/A se(a) 14 R 98.1% 42.8 s 15 F 358.5 7 df 16 ssreg 658159 12851 ssresid 17 18 risk notation AY 2008 Formula 19 X 21,234 20 Y 63,281 bX =B12*C19 21 parameter Δ(Y) 3,342.10 X∙se(b ) =C19*B13 22 process Γ(Y) 6,243.48 sqrt(X)∙s =sqrt(C19)*C14 23 total se(Y) 7,081.71 sqrt(Δ2+Γ2) =sqrt(B21^2+B22^2)

Remove heteroscedasticity inside LINEST with array version of SQRT

• Δ: Parameter risk = variability in estimate of expected value
• Γ: Process risk = variability due to all other factors not explained by X

LINEST

• utput

CAS Annual Meeting 2009

16

AY2008 12-mo value

{=LINEST(C2:C9/SQRT(B2:B9),SQRT(B2:B9), FALSE,TRUE)}

z X b X Y  + = /

• Estimates for b, σ

will be same in both models

z X bX Y  + =

becomes

“Don’t give me an intercept, but give me all the yummy statistics!”

A B C D 1 AY \ Age 1 2 3 2 2000 10,238 24,654 38,025 3 2001 5,508 16,235 25,586 4 2002 7,374 20,620 34,220 5 2003 6,153 19,182 31,005 6 2004 7,253 25,066 40,134 7 2005 10,855 38,520 62,348 8 2006 10,313 34,341 51,110 9 2007 16,411 42,228 66,770 10 2008 21,234 63,281 100,059 11 12 b 2 1.581 a 13 se(b 2 ) 0.023 #N/A se(a) 14 R 100% 9.6 s 2 15 F 4888.3 6 df 16 ssreg 446571 548 ssresid 17

risk notation AY 2008 Formula

18 Y 1 63,281 19

Y 2 100,059 b 2 Y 1

20 parameter Δ(Y 2 )

5,475.36

sqrt( Y 1

2*se(b 2)2 + b2 2*Δ(Y 1)2 + se(b 2)2*Δ(Y 1)2 )

21

process

Γ(Y 2 )

10,324.69

sqrt( Y 1*s 2

2 + b 2 2*Γ(Y 1)2 )

22

total

se(Y 2)

11,686.69

sqrt(Δ2+Γ2)

Second development period: chained formulas for errors more complicated than for expected values

• Formulas relatively easy to copy cell to cell

CAS Annual Meeting 2009

17

• Errors are compounded when

beginning value Y1 is estimated

• Use LINEST to find b, s for

second development period

• Error formulas
• For 2007: same as before
• For 2008: more formidable

error Y b Y + =

1 , 2008 2 2 , 2008

{=LINEST(D2:D8/SQRT(C2:C8),SQRT(C2:C8), FALSE,TRUE)}

ABC Insurance Company Chain Ladder Loss Projection

AY \ Age 1 2 3 4 5 6 7 8 9 = Ult 2000 10,238 24,654 38,025 46,550 52,842 58,722 65,227 67,604 69,559 2001 5,508 16,235 25,586 32,863 38,111 42,315 45,171 47,666 49,045 2002 7,374 20,620 34,220 43,438 50,898 55,475 58,367 60,943 62,706 2003 6,153 19,182 31,005 40,424 46,949 50,942 54,931 57,354 59,014 2004 7,253 25,066 40,134 51,063 58,376 64,144 69,166 72,218 74,307 2005 10,855 38,520 62,348 82,710 95,382 104,806 113,011 117,998 121,411 2006 10,313 34,341 51,110 65,632 75,688 83,166 89,677 93,634 96,343 2007 16,411 42,228 66,770 85,743 98,879 108,649 117,155 122,324 125,863 2008 21,234 63,281 100,059 128,491 148,177 162,818 175,564 183,311 188,614 Sum of unpaid loss 63,281 166,829 279,866 418,125 523,583 619,504 707,782

777,301

• pt. est.

= (42228+63281)*1.581

33,566

total risk

Error formulas for AY sum

• f unpaid loss are similar – refer to papers

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 700000 720000 740000 760000 780000 800000 820000 840000 860000 880000

Fitted Lognormal

• Lognormal parameters

(method of moments): μ=13.6, σ=.04

• Use fitted distribution

for risk inferences

CAS Annual Meeting 2009

18

Same as for AY 2008 alone Can fit any 2- parameter distribution to first 2 moments of estimated unpaid loss Uses “Y1” = 42,228 + 63,281 and Δ, Γ from age 2 sum

Simple average link ratios are the optimal solution

• f a model with a different variance assumption
• Divide both sides by X to get OLS model with constant variance
• {=LINEST(C2:C9/B2:B9,B2:B9/B2:B9,FALSE,TRUE)}

A B C D E F G 1 AY \ Age 1 2 ata 2 2000 10,238 24,654 2.408 3 2001 5,508 16,235 2.948 4 2002 7,374 20,620 2.796 5 2003 6,153 19,182 3.118 6 2004 7,253 25,066 3.456 7 2005 10,855 38,520 3.549 8 2006 10,313 34,341 3.330 9 2007 16,411 42,228 2.573 10 3.022 simple average 11 12 b 3.022 a 13 se(b ) 0.147 #N/A se(a) 14 R 98.4% 0.4 s 15 F 424.7 7 df 16 ssreg 73.066 1.204 ssresid 17 18 risk notation AY 2008 Formula 19 X 21,234 20 Y 1 64,172 bX 21 parameter Δ(Y 1 ) 3,113.93 X∙se(b ) 22 process Γ(Y 1 ) 60.44 sqrt(X)∙s 23 total se(Y1) 3,114.52 sqrt(Δ2+Γ2)

z X X b Y

1 1 1

 + =

CAS Annual Meeting 2009

19

Average-x-high-low link ratio is optimal solution

• f a model with a different variance assumption
• See eForum paper Bardis, Majidi, Murphy

A B C D E F G 1 AY \ Age 1 2 ata 2 2000 10,238 24,654 2.408 3 2001 5,508 16,235 2.948 4 2002 7,374 20,620 2.796 5 2003 6,153 19,182 3.118 6 2004 7,253 25,066 3.456 7 2005 10,855 38,520 3.549 8 2006 10,313 34,341 3.330 9 2007 16,411 42,228 2.573 10 3.037 average x hi-lo 11 α 12 b 3.037 a 2.54891151 13 se(b ) 0.139 #N/A se(a) 14 R 98.55% 0.0 s 15 F 476.5 7 df 16 ssreg 0.5146 0.0076 ssresid 17 18 risk notation AY 2008 Formula 19 X 21,234 20 Y 1 64,485 bX 21 parameter Δ(Y 1 ) 2,954.13 X∙se(b ) 22 process Γ(Y 1 ) 4.79 sqrt(X)∙s 23 total se(Y1) 2,954.14 sqrt(Δ2+Γ2)

z X X b Y

1 2 1 1





+ =

α Type of ata 2 Simple average 1 Weighted average linear regression

• ther

Actuarial selection

CAS Annual Meeting 2009

20

Many selected link ratios – not necessarily all – can be optimal within this family of α-indexed models

• Given triangle data over a development period, reasonable link

ratios can be viewed as LINEST solutions for some index α

• Use Excel’s “What-If” analysis to generate above graph from your
• wn triangle, “Goal Seek” to find α given your selection

z X bX Y 

 2

+ =

2.8 2.85 2.9 2.95 3 3.05 3.1

• 1

1 2 3 4 5 6 ata factor alpha

Reasonable Link Ratio Function Age 1-2

CAS Annual Meeting 2009

21

Trinostics LLC is in the business of collaboration and education in the design and construction

• f transparently valuable

actuarial models Daniel Murphy, FCAS, MAAA dmurphy@trinostics.com

CAS Annual Meeting 2009

22